Laplacian (Real Scalar Fields)#
Definition: Laplacian
Let \(f: \mathcal{D} \subseteq \mathbb{R}^n \to \mathbb{R}\) be a real scalar field which is twice totally differentiable at \(\boldsymbol{p} \in \operatorname{int} \mathcal{D}\).
The Laplacian of \(f\) at \(\boldsymbol{p}\) is the divergence of its gradient there:
\[\nabla \cdot \nabla f(\boldsymbol{p})\]
Notation
\[\nabla^2 f (\boldsymbol{p}) \qquad \Delta f(\boldsymbol{p})\]
Example: \(f(x,y) = \sin(\uppi x)\sin (\uppi y)\)
Consider the real scalar field \(f: \mathbb{R}^2 \to \mathbb{R}\) defined as follows:
\[f(x, y) = \sin(\uppi x)\sin (\uppi y)\]
We see that \(f\) is twice totally differentiable on \(\mathbb{R}^2\) and its Laplacian is given by its partial derivatives as follows:
\[\begin{aligned}\nabla^2 f(x,y) & = \nabla \cdot \nabla f(x,y) \\ & = \operatorname{div} \begin{bmatrix} \uppi \cos (\uppi x) \sin(\uppi y) \\ \uppi \sin (\uppi x) \cos (\uppi y)\end{bmatrix} \\ & = \partial_x (\uppi \cos (\uppi x) \sin(\uppi y)) + \partial_y (\uppi \sin (\uppi x) \cos (\uppi y)) \\ & = - \uppi^2 \sin(\uppi x)\sin (\uppi y) - \uppi^2 \sin(\uppi x)\sin (\uppi y) \\ & = -2 \uppi^2 \sin(\uppi x)\sin (\uppi y)\end{aligned}\]
Theorem: Green's First Identity
\[\iiint_{\mathcal{D}} (\nabla^2 f) g \, \mathrm{d}\mathcal{D} = -\iiint_{\mathcal{D}} \nabla f \cdot \nabla g \, \mathrm{d}\mathcal{D} + \iint_{\partial \mathcal{D}} g \nabla f \cdot \mathbf{n} \, \mathrm{d}S\]
Proof
TODO